If you’re a guitarist, you may have noticed that it’s hard to get your instrument perfectly in tune. This is not your imagination. If you tune each string perfectly to the one next to it, the low E string will end up out of tune with the high E string. If you use an electronic tuner to make sure the individual strings are tuned to the correct pitch, they won’t sound fully in tune with each other. It has nothing to do with the quality of your instrument or your skill at tuning: it’s a fundamental fact of western music theory. This post attempts to explain why. It’s very geeky stuff, but if you like math (and who doesn’t?) then read on.

To understand tuning, it helps to start with the concept of the octave. Two pitches are an octave apart if their frequencies have the ratio 2:1. Standard concert A has a frequency of 440 Hz. When you play concert A on the guitar, the string vibrates to and fro 440 times every second. If you double the frequency to 880 Hz, you get an A that’s one octave higher. If you halve the frequency to 220 Hz, you get an A that’s an octave lower. The ear hears all these different pitches as being the “same” note. (Technically, they’re the same pitch class.) This ability we have to hear frequencies related by powers of two as being the “same” is known in music theory terms as octave equivalency. This ability isn’t specific to humans. Rhesus monkeys hear octaves as being equivalent too.

Octaves emerge naturally out of the overtone series. The first harmonic of a vibrating string is an octave above the fundamental. The third harmonic is two octaves above. The seventh harmonic is three octaves above, and the fifteenth harmonic is four octaves above.

After the octave, the next musical interval you get from the natural overtone series is the fifth (it’s the third harmonic.) Two pitches are a fifth apart if the ratio between their frequencies is 3:2. The note a fifth above concert A (440 Hz) is E (660 Hz.)

Fifths are a very significant interval in western music theory. If you keep going up by fifths, you visit every note in the chromatic scale (every key on the piano) until you eventually wind up back on the note where you began. So if you start on A, then go up to E, then B, then F#, and so on, eventually you’ll wind up on the A seven octaves higher from where you started. This concept is known as the circle of fifths, though it would be more accurate to call it the spiral of fifths, since you’re getting higher and higher in pitch. The circle of fifths is foundational to western music theory. It makes it possible to transpose music effortlessly from one key to another. The circle gives rise to all sorts of useful and interesting symmetries, too, like its close relationship to the circle of semitones.

But there’s a big problem with the circle of fifths. If you use the 3/2 ratio you get from the natural overtone series, the circle doesn’t actually close. Recall that to go up by a fifth, you multiply the frequency by 3/2. To keep going up by fifths, you keep multiplying by 3/2. To go all the way around the circle of fifths from A to A, you multiply by 3/2 twelve times:

(3/2)^12 = 129.746337890625

Going around the circle of fifths twelve times is the same as going up seven octaves. To go up an octave, you multiply by two, so to go up seven octaves, you multiply by two seven times:

2^7 = 128

Going from A to A by fifths means multiplying the frequency by 129.746337890625, but going by octaves means multiplying by 128. The discrepancy between the two multiples is known in music theory terms as the Pythagorean comma, and it has caused musicians a lot of gray hair over the past few hundred years. It would be nice if a tuning system based on fifths agreed with a system based on octaves. It would make it a lot easier to hop from one key to another without having to retune your instruments. But that is sadly not possible.

The history of western tuning systems is the story of musicians trying to resolve the contradiction between the desire to have pure-sounding overtone-based intervals and a closed circle of fifths. European musicians of the 1700s tried all kinds of compromises. You could have some of the keys sound perfectly in tune, and have others be out of tune. You could have eleven decent-sounding keys and one awful one. You could use perfect fifths and smooth out the Pythagorean comma with out-of-tune thirds. You could have pianos with many extra keys for all the subtly different versions of each note.

I don’t advise getting too bogged down in the minutiae of all these different systems. The bottom line is that the western world eventually settled on its present consensus solution, which is to just make all the intervals other than octaves a little bit wrong. This system is called equal temperament. It’s considered a “modern” idea, but it dates back at least as far as Galileo’s father Vincenzo Galilei.

In equal temperament, all intervals are built by adding semitones together, and all semitones are defined as a ratio of one to the twelfth root of two. Twelve half steps gets you the perfect octave, because multiplying by the twelfth root of two twelve times equals two. An equal-tempered fifth is seven semitones — you multiply the frequency by 2^(7/12). This comes to about 1.4983, which isn’t quite the 3/2 from the overtone system that your ear would like, but it’s close enough to not be offensively awful-sounding. The other equal-tempered intervals are similarly “wrong,” but by similarly bearable small amounts. Every key is identical and the circle of fifths closes, so everybody is more or less happy. If you get an electronic guitar tuner, it’ll be based on equal temperament.

Some musicians lament the loss of pure fifths. One bassist I know claims that all those out-of-tune fifths are gradually making western listeners crazy, which is why we’ve had so many enormous and horrible wars in the past couple of centuries. This idea sounds silly to me, but it’s true that pure fifths are easier on the brain. On instruments where the tuning is flexible, like winds and violin, the most skilled musicians tend to seek out pure intervals by ear, adjusting their intonation slightly depending on the key. Good singers do this too. Electronic instruments are a lot easier to retune than acoustic ones, and it’s sometimes possible to program in whatever tuning system suits you. Auto-tune lets you choose any historical or microtonal tuning system you want right off a menu.

So what if you’re just trying to get your guitar in tune? You need to make peace with not being able to do it perfectly. Use an electronic tuner to get the individual strings to their correct equal-tempered pitches and deal with the fifths sounding a little wrong, or tune with harmonics and have the low register not quite match the high register. In practice, most guitarists just fudge a little bit one way or the other, and guitars rarely stay tuned the way you want them to anyway. As always, let your ear be your guide.

Fascinating stuff Ethan. I Happened upon your blog trying to find your writings about the democratisation of music and the idea that the specialisation that the record industry gave is a cultural and historical anomaly, all for my MA in Music Industries – http://mamusicind.posterous.com – … Happened upon this and I’ll be checking my tuning tonight.

😉

Stay well, keep fighting the good fight… and death to false metal

Wow! So happy to have stumbled upon your site, and I will peruse with enthusiasm in a minute. But first, I wanted to share a bit of my blog with you, also pertaining to guitar tuning. Here an older article called “Totally Flawsome”:

http://grahamophonesbadguitarchannel.blogspot.com/2010/05/gbgc-7-heart-of-glass-its-all-coming.html

Thanks for stumbling. I dug your post, I agree there’s something endearing about an instrument that’s “flawsome.” When I’m doing electronic music I like to find samples that are a little “wrong” to keep the whole thing from being too sterile. It’s interesting to me that there’s a whole genre of electronica, dubstep, where the whole point is to have beats and synth parts that are intentionally programmed to be a little off.

Language correction:

Sorry, I am Danish!

Please substitute ‘derive’ for ‘deviate from’ in my comment above!

Glad you enjoyed. I know it’s weird to divide the spiral of fifths into fours that way but it’s the best 3D image of a spiral that I have. If you can recommend a better one, I’ll happily use it instead. The video is useful for making all the math more tangible, check it out folks.

Great article, thanks! I am also very happy about the link to articles about neurological research.

Somewhat less happy about your representation of the spiral of fifths as it only makes sense to divide each turn in four from a limited perspective.

Maybe my video about the pythagorean comma will elucidate some of the facts you are describing. Please note that in the video the piano keyboard is only to make the structure recognizable for those who have difficulties grasping the abstract idea of the octave/fifth spiral.

http://www.youtube.com/watch?v=Au2QFQ2ppyg

The problem is of course that a piano layout gives you the illusion of tonal fix points whereas the whole idea – like you described it so clearly – is that the tonal frequencies differ according to tuning system. For example h# generated by 12 pure fifths is about a quarter of a half tone (pyt. comma) higher than the c we set off from.

Another important consideration is that the equal temperament is much, much harder on the pure thirds and sixths than the pure fifths. The later only derive by about 2 cents (one fiftieth of a half tone) whereas the former (and there are four of them – minor and major third and sixth) derive by about 15 cents – one seventh of a half tone!

The middle tone temperament which was much used during the renaissance was designed to keep the thirds pure.

Cheers!

Very cool article. I’d known this since reading Issacoff’s book on temperament, but somehow hadn’t made the jump to my own guitar.